<p>A fundamental result by Kemer states that there are only two varieties of associative algebras with almost polynomial growth. When we consider associative algebras with additional structure, several other varieties with almost polynomial growth arise. In this survey paper, we give an overview of the classification theorems of such varieties. We address the topic in a unified manner using universal algebra. Moreover, we extend some results in the context of graded maps of superalgebras and we provide a new example of a differential variety with almost polynomial growth.</p>